Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,29)(27,30)(31,32);(1,2,4,3)(9,11,12,15)(10,13,14,16)(19,26)(22,29)(27,32)(30,31);(10,14)(13,16)(17,19,21,26)(18,22,25,29)(20,27,28,32)(23,30,24,31);(1,3,4,2)(18,23)(20,28)(22,30)(24,25)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,g^d=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264349, #72264355, #72264386, … |
| maximal normal subgroup of | #72264349, #72264355, #72264386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299596);